Section 3.6One-To-One Functions; Inverse Functions

Section 3.6One-to-one Functions; Inverse Functions

Objective 1: Understanding the Definition of a One-to-one Function

DefinitionOne-to-one Function

A function is one-to-one if for any values in the domain of , .

Interpretation: For a function f(x) = y, we know that for each x in the Domain there exists one and only one y in the Range. For a one-to-one function f(x) = y, for each x in the Domain there exists one and only one y in the Range AND for each y in the Range there exists one and only one x in the Domain.

Objective 2: Determining if a Function is One-to-one Using the Horizontal Line Test

The Horizontal Line TestIf every horizontal line intersects the graph of a function at most once, then

is one-to-one.

Objective 3: Understanding and Verifying Inverse Functions

Every one-to-one function has an inverse function.

DefinitionInverse Function

Let be a one-to-one function with domain A and range B. Then is the inverse

function of with domain B and range A. Furthermore, if then .

Domain of Range of

Range of Domain of

•a •b

Do not confuse with . The negative 1 in is NOT an exponent!

Inverse functions “undo” each other.

Composition Cancellation Equations

for all in the domain of and

for all in the domain of

Objective 4: Sketching the Graphs of Inverse Functions

The graph of is a reflection of the graph of about the line .

If the functions have any points in common, they must lie along the line .

The graph ofThe graph of

a one-to-oneand

function and

its inverse.

Objective 5: Finding the Inverse of a One-to-one Function

We know that if a point is on the graph of a one-to-one function, then the point is on the graph of its inverse function.

To find the inverse of a one-to-one function, replace with y, interchange the variables then solve for y. This is the function .

Inverse Function Summary

1.The inverse function exists if and only if the function is one-to-one.

2. The domain of is the same as the range of and the range of

is the same as the domain of .

3. To verify that two one-to-one functions are inverses of each other, use

the composition cancellation equations to show that .

4.The graph of is a reflection of the graph of about the line . That is, for any

point that lies on the graph of f, the point must lie on the graph of .

5.To find the inverse of a one-to-one function, replace with y, interchange the

variables then solve for y. This is the function .